1 /** 2 * \file AlbersEqualArea.hpp 3 * \brief Header for GeographicLib::AlbersEqualArea class 4 * 5 * Copyright (c) Charles Karney (2010-2021) <charles@karney.com> and licensed 6 * under the MIT/X11 License. For more information, see 7 * https://geographiclib.sourceforge.io/ 8 **********************************************************************/ 9 10 #if !defined(GEOGRAPHICLIB_ALBERSEQUALAREA_HPP) 11 #define GEOGRAPHICLIB_ALBERSEQUALAREA_HPP 1 12 13 #include <GeographicLib/Constants.hpp> 14 15 namespace GeographicLib { 16 17 /** 18 * \brief Albers equal area conic projection 19 * 20 * Implementation taken from the report, 21 * - J. P. Snyder, 22 * <a href="http://pubs.er.usgs.gov/usgspubs/pp/pp1395"> Map Projections: A 23 * Working Manual</a>, USGS Professional Paper 1395 (1987), 24 * pp. 101--102. 25 * 26 * This is a implementation of the equations in Snyder except that divided 27 * differences will be [have been] used to transform the expressions into 28 * ones which may be evaluated accurately. [In this implementation, the 29 * projection correctly becomes the cylindrical equal area or the azimuthal 30 * equal area projection when the standard latitude is the equator or a 31 * pole.] 32 * 33 * The ellipsoid parameters, the standard parallels, and the scale on the 34 * standard parallels are set in the constructor. Internally, the case with 35 * two standard parallels is converted into a single standard parallel, the 36 * latitude of minimum azimuthal scale, with an azimuthal scale specified on 37 * this parallel. This latitude is also used as the latitude of origin which 38 * is returned by AlbersEqualArea::OriginLatitude. The azimuthal scale on 39 * the latitude of origin is given by AlbersEqualArea::CentralScale. The 40 * case with two standard parallels at opposite poles is singular and is 41 * disallowed. The central meridian (which is a trivial shift of the 42 * longitude) is specified as the \e lon0 argument of the 43 * AlbersEqualArea::Forward and AlbersEqualArea::Reverse functions. 44 * AlbersEqualArea::Forward and AlbersEqualArea::Reverse also return the 45 * meridian convergence, γ, and azimuthal scale, \e k. A small square 46 * aligned with the cardinal directions is projected to a rectangle with 47 * dimensions \e k (in the E-W direction) and 1/\e k (in the N-S direction). 48 * The E-W sides of the rectangle are oriented γ degrees 49 * counter-clockwise from the \e x axis. There is no provision in this class 50 * for specifying a false easting or false northing or a different latitude 51 * of origin. 52 * 53 * Example of use: 54 * \include example-AlbersEqualArea.cpp 55 * 56 * <a href="ConicProj.1.html">ConicProj</a> is a command-line utility 57 * providing access to the functionality of LambertConformalConic and 58 * AlbersEqualArea. 59 **********************************************************************/ 60 class GEOGRAPHICLIB_EXPORT AlbersEqualArea { 61 private: 62 typedef Math::real real; 63 real eps_, epsx_, epsx2_, tol_, tol0_; 64 real _a, _f, _fm, _e2, _e, _e2m, _qZ, _qx; 65 real _sign, _lat0, _k0; 66 real _n0, _m02, _nrho0, _k2, _txi0, _scxi0, _sxi0; 67 static const int numit_ = 5; // Newton iterations in Reverse 68 static const int numit0_ = 20; // Newton iterations in Init hyp(real x)69 static real hyp(real x) { 70 using std::hypot; 71 return hypot(real(1), x); 72 } 73 // atanh( e * x)/ e if f > 0 74 // atan (sqrt(-e2) * x)/sqrt(-e2) if f < 0 75 // x if f = 0 atanhee(real x) const76 real atanhee(real x) const { 77 using std::atan; using std::abs; using std::atanh; 78 return _f > 0 ? atanh(_e * x)/_e : (_f < 0 ? (atan(_e * x)/_e) : x); 79 } 80 // return atanh(sqrt(x))/sqrt(x) - 1, accurate for small x 81 static real atanhxm1(real x); 82 83 // Divided differences 84 // Definition: Df(x,y) = (f(x)-f(y))/(x-y) 85 // See: 86 // W. M. Kahan and R. J. Fateman, 87 // Symbolic computation of divided differences, 88 // SIGSAM Bull. 33(3), 7-28 (1999) 89 // https://doi.org/10.1145/334714.334716 90 // http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf 91 // 92 // General rules 93 // h(x) = f(g(x)): Dh(x,y) = Df(g(x),g(y))*Dg(x,y) 94 // h(x) = f(x)*g(x): 95 // Dh(x,y) = Df(x,y)*g(x) + Dg(x,y)*f(y) 96 // = Df(x,y)*g(y) + Dg(x,y)*f(x) 97 // = Df(x,y)*(g(x)+g(y))/2 + Dg(x,y)*(f(x)+f(y))/2 98 // 99 // sn(x) = x/sqrt(1+x^2): Dsn(x,y) = (x+y)/((sn(x)+sn(y))*(1+x^2)*(1+y^2)) Dsn(real x,real y,real sx,real sy)100 static real Dsn(real x, real y, real sx, real sy) { 101 // sx = x/hyp(x) 102 real t = x * y; 103 return t > 0 ? (x + y) * Math::sq( (sx * sy)/t ) / (sx + sy) : 104 (x - y != 0 ? (sx - sy) / (x - y) : 1); 105 } 106 // Datanhee(x,y) = (atanee(x)-atanee(y))/(x-y) 107 // = atanhee((x-y)/(1-e^2*x*y))/(x-y) Datanhee(real x,real y) const108 real Datanhee(real x, real y) const { 109 real t = x - y, d = 1 - _e2 * x * y; 110 return t == 0 ? 1 / d : 111 (x*y < 0 ? atanhee(x) - atanhee(y) : atanhee(t / d)) / t; 112 } 113 // DDatanhee(x,y) = (Datanhee(1,y) - Datanhee(1,x))/(y-x) 114 real DDatanhee(real x, real y) const; 115 real DDatanhee0(real x, real y) const; 116 real DDatanhee1(real x, real y) const; 117 real DDatanhee2(real x, real y) const; 118 void Init(real sphi1, real cphi1, real sphi2, real cphi2, real k1); 119 real txif(real tphi) const; 120 real tphif(real txi) const; 121 122 friend class Ellipsoid; // For access to txif, tphif, etc. 123 public: 124 125 /** 126 * Constructor with a single standard parallel. 127 * 128 * @param[in] a equatorial radius of ellipsoid (meters). 129 * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. 130 * Negative \e f gives a prolate ellipsoid. 131 * @param[in] stdlat standard parallel (degrees), the circle of tangency. 132 * @param[in] k0 azimuthal scale on the standard parallel. 133 * @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k0 is 134 * not positive. 135 * @exception GeographicErr if \e stdlat is not in [−90°, 136 * 90°]. 137 **********************************************************************/ 138 AlbersEqualArea(real a, real f, real stdlat, real k0); 139 140 /** 141 * Constructor with two standard parallels. 142 * 143 * @param[in] a equatorial radius of ellipsoid (meters). 144 * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. 145 * Negative \e f gives a prolate ellipsoid. 146 * @param[in] stdlat1 first standard parallel (degrees). 147 * @param[in] stdlat2 second standard parallel (degrees). 148 * @param[in] k1 azimuthal scale on the standard parallels. 149 * @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k1 is 150 * not positive. 151 * @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in 152 * [−90°, 90°], or if \e stdlat1 and \e stdlat2 are 153 * opposite poles. 154 **********************************************************************/ 155 AlbersEqualArea(real a, real f, real stdlat1, real stdlat2, real k1); 156 157 /** 158 * Constructor with two standard parallels specified by sines and cosines. 159 * 160 * @param[in] a equatorial radius of ellipsoid (meters). 161 * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. 162 * Negative \e f gives a prolate ellipsoid. 163 * @param[in] sinlat1 sine of first standard parallel. 164 * @param[in] coslat1 cosine of first standard parallel. 165 * @param[in] sinlat2 sine of second standard parallel. 166 * @param[in] coslat2 cosine of second standard parallel. 167 * @param[in] k1 azimuthal scale on the standard parallels. 168 * @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k1 is 169 * not positive. 170 * @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in 171 * [−90°, 90°], or if \e stdlat1 and \e stdlat2 are 172 * opposite poles. 173 * 174 * This allows parallels close to the poles to be specified accurately. 175 * This routine computes the latitude of origin and the azimuthal scale at 176 * this latitude. If \e dlat = abs(\e lat2 − \e lat1) ≤ 160°, 177 * then the error in the latitude of origin is less than 4.5 × 178 * 10<sup>−14</sup>d;. 179 **********************************************************************/ 180 AlbersEqualArea(real a, real f, 181 real sinlat1, real coslat1, 182 real sinlat2, real coslat2, 183 real k1); 184 185 /** 186 * Set the azimuthal scale for the projection. 187 * 188 * @param[in] lat (degrees). 189 * @param[in] k azimuthal scale at latitude \e lat (default 1). 190 * @exception GeographicErr \e k is not positive. 191 * @exception GeographicErr if \e lat is not in (−90°, 192 * 90°). 193 * 194 * This allows a "latitude of conformality" to be specified. 195 **********************************************************************/ 196 void SetScale(real lat, real k = real(1)); 197 198 /** 199 * Forward projection, from geographic to Lambert conformal conic. 200 * 201 * @param[in] lon0 central meridian longitude (degrees). 202 * @param[in] lat latitude of point (degrees). 203 * @param[in] lon longitude of point (degrees). 204 * @param[out] x easting of point (meters). 205 * @param[out] y northing of point (meters). 206 * @param[out] gamma meridian convergence at point (degrees). 207 * @param[out] k azimuthal scale of projection at point; the radial 208 * scale is the 1/\e k. 209 * 210 * The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No 211 * false easting or northing is added and \e lat should be in the range 212 * [−90°, 90°]. The values of \e x and \e y returned for 213 * points which project to infinity (i.e., one or both of the poles) will 214 * be large but finite. 215 **********************************************************************/ 216 void Forward(real lon0, real lat, real lon, 217 real& x, real& y, real& gamma, real& k) const; 218 219 /** 220 * Reverse projection, from Lambert conformal conic to geographic. 221 * 222 * @param[in] lon0 central meridian longitude (degrees). 223 * @param[in] x easting of point (meters). 224 * @param[in] y northing of point (meters). 225 * @param[out] lat latitude of point (degrees). 226 * @param[out] lon longitude of point (degrees). 227 * @param[out] gamma meridian convergence at point (degrees). 228 * @param[out] k azimuthal scale of projection at point; the radial 229 * scale is the 1/\e k. 230 * 231 * The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No 232 * false easting or northing is added. The value of \e lon returned is in 233 * the range [−180°, 180°]. The value of \e lat returned is 234 * in the range [−90°, 90°]. If the input point is outside 235 * the legal projected space the nearest pole is returned. 236 **********************************************************************/ 237 void Reverse(real lon0, real x, real y, 238 real& lat, real& lon, real& gamma, real& k) const; 239 240 /** 241 * AlbersEqualArea::Forward without returning the convergence and 242 * scale. 243 **********************************************************************/ Forward(real lon0,real lat,real lon,real & x,real & y) const244 void Forward(real lon0, real lat, real lon, 245 real& x, real& y) const { 246 real gamma, k; 247 Forward(lon0, lat, lon, x, y, gamma, k); 248 } 249 250 /** 251 * AlbersEqualArea::Reverse without returning the convergence and 252 * scale. 253 **********************************************************************/ Reverse(real lon0,real x,real y,real & lat,real & lon) const254 void Reverse(real lon0, real x, real y, 255 real& lat, real& lon) const { 256 real gamma, k; 257 Reverse(lon0, x, y, lat, lon, gamma, k); 258 } 259 260 /** \name Inspector functions 261 **********************************************************************/ 262 ///@{ 263 /** 264 * @return \e a the equatorial radius of the ellipsoid (meters). This is 265 * the value used in the constructor. 266 **********************************************************************/ EquatorialRadius() const267 Math::real EquatorialRadius() const { return _a; } 268 269 /** 270 * @return \e f the flattening of the ellipsoid. This is the value used in 271 * the constructor. 272 **********************************************************************/ Flattening() const273 Math::real Flattening() const { return _f; } 274 275 /** 276 * @return latitude of the origin for the projection (degrees). 277 * 278 * This is the latitude of minimum azimuthal scale and equals the \e stdlat 279 * in the 1-parallel constructor and lies between \e stdlat1 and \e stdlat2 280 * in the 2-parallel constructors. 281 **********************************************************************/ OriginLatitude() const282 Math::real OriginLatitude() const { return _lat0; } 283 284 /** 285 * @return central scale for the projection. This is the azimuthal scale 286 * on the latitude of origin. 287 **********************************************************************/ CentralScale() const288 Math::real CentralScale() const { return _k0; } 289 290 /** 291 * \deprecated An old name for EquatorialRadius(). 292 **********************************************************************/ 293 GEOGRAPHICLIB_DEPRECATED("Use EquatorialRadius()") MajorRadius() const294 Math::real MajorRadius() const { return EquatorialRadius(); } 295 ///@} 296 297 /** 298 * A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e 299 * stdlat = 0, and \e k0 = 1. This degenerates to the cylindrical equal 300 * area projection. 301 **********************************************************************/ 302 static const AlbersEqualArea& CylindricalEqualArea(); 303 304 /** 305 * A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e 306 * stdlat = 90°, and \e k0 = 1. This degenerates to the 307 * Lambert azimuthal equal area projection. 308 **********************************************************************/ 309 static const AlbersEqualArea& AzimuthalEqualAreaNorth(); 310 311 /** 312 * A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e 313 * stdlat = −90°, and \e k0 = 1. This degenerates to the 314 * Lambert azimuthal equal area projection. 315 **********************************************************************/ 316 static const AlbersEqualArea& AzimuthalEqualAreaSouth(); 317 }; 318 319 } // namespace GeographicLib 320 321 #endif // GEOGRAPHICLIB_ALBERSEQUALAREA_HPP 322